1. Field of the Invention
The present invention relates to the field of optics. More particularly, the present invention relates to an optical system that transforms a beam having a substantially non-uniform intensity profile to a beam whose intensity profile is substantially uniform.
2. Description of the Related Art
Lasers emitting collimated beams of coherent light have many applications in optical science and technology, including lithography, spectroscopy, communications and display technology. Due to fundamental properties of light propagation in optical resonators, most lasers emit beams having a light intensity that is extremely inhomogeneous. Specifically, the light intensity of an emitted laser beam generally follows a Gaussian distribution
I(r)=(2P/xcfx80w2)exe2x88x922r2/w2,xe2x80x83xe2x80x83(1)
where I(r) denotes the optical power per unit area measured at a distance r from the axis of the beam, P denotes the total power of the beam, and w is the beam waist parameter, which sets the length scale over which the optical intensity declines from its maximum value to zero. The same distribution also describes, to a good approximation, the intensity profile of a beam that emerges from a single-mode optical fiber, such as is used extensively in the optical industry for conveying light.
For many applications, it is desirable that some area of interest be illuminated as uniformly as possible. For example, optical lithography, which is used to fabricate microelectronic devices, requires that the light fluence over an entire exposed region conform to tight tolerances. Laser users, therefore, frequently encounter the problem of transforming a beam having a Gaussian intensity profile to an output beam having optical intensity that is substantially uniform, e.g., a so-called flat-top profile having uniform intensity over a circular or rectangular region.
Many solutions have been proposed for transforming a Gaussian beam to a flat-top beam. All conventional solutions, however, have significant drawbacks. For example, the conceptually-simplest conventional method uses an element having radially-varying absorption for removing excess intensity from the center of a beam. Such an approach is inherently inefficient because it can be shown that, in the best case, the fraction of the incident beam power that emerges in the beam is 1/e, or approximately 37%. Moreover, in this conventional approach, the absorptive element only subtends the central part of the incoming beam, and has an aperture or other discontinuity located at a point where the light intensity is an appreciable fraction of the peak intensity. When using spatially coherent light sources, including most lasers, any aperture that truncates the beam also diffracts light into the central region of the beam. Accordingly, interference of the diffracted and transmitted light reduces the uniformity of the beam. Yet another drawback of this conventional approach is that stable, well-characterized absorptive materials are required, which are not available for the technologically-important ultraviolet wavelengths.
Another conventional approach uses lithographically or holographically fabricated phase gratings for reshaping a Gaussian beam by diffraction. Holographic gratings suffer from limited diffraction efficiency of only about 30%, as well as a lack of materials that are suitable for ultraviolet applications. Lithographically-fabricated phase gratings can have high efficiency, but are expensive to fabricate and only work as designed for a single wavelength. Additionally, it is exceedingly difficult to avoid diffraction into unwanted orders, leading to undesirable effects, such as non-uniformity of the output beam at high spatial frequencies and xe2x80x9chot spotsxe2x80x9d on the beam axis.
Conventional refractive solutions have been proposed that use either spherical or aspheric optical elements for aberrating and then recollimating a laser beam. The solutions with conventional, spherical optics are physically bulky and relatively inefficient because the spherical surfaces introduce limited aberrations. More compact and efficient conventional designs have been proposed that use either aspheric or gradient-index lenses. Nevertheless, use of a gradient index accomplishes essentially the same result as an aspheric surface, but with the drawback that no gradient-index glasses are available for ultraviolet applications. Prior art aspheric and gradient-index solutions have favored the use of a negative first element and a positive second element in a configuration resembling a Galilean telescope. Unfortunately, such designs require lenses that are difficult to fabricate. Fabrication problems are especially acute for the concave surface of the first lens in the Galilean design.
Another serious problem commonly encountered in the prior art is that the solutions to the aforementioned problem are generally only valid for the central region of an incident beam, thereby entailing an aperture or other discontinuity at a point where there is appreciable input beam intensity, on the order of (1/e)2 times the peak intensity. As previously mentioned, truncating an input beam causes diffraction and interference fringes that reduce the uniformity of the output beam.
There is still a need to efficiently transform a non-uniform beam to a beam having uniform optical intensity.
In one embodiment of the present invention, there is provided an optical system that transforms a light beam having an axially-symmetric intensity distribution to a light beam having another axially-symmetric intensity distribution.
In a preferred embodiment of the invention, there is provided an optical system that comprises at least one optical element that includes a first and a second aspherical, non-planar, non-reentrant surface, in which the first and second surfaces are aligned along an optical axis and are configured to transform substantially all of an input beam incident on the first surface to an output beam propagating away from the second surface. The intensity profile of the input beam is expressible as a first function of a first coordinate times a second function of a second coordinate, in which the first and the second coordinates are independent of each other. The intensity profile of the output beam has a shape that is substantially different from that of the input beam, and the output beam has an intensity profile that is sigmoidal to reduce diffraction effects. In a preferred embodiment, the optical system comprises first and second optical elements that include the first and the second surfaces, respectively. Preferred embodiments are directed to either a Keplerian or a Galilean configuration of the surfaces. In a preferred embodiment, substantially all of the input beam is transformed for any wavelength within the wavelength range extending from 257 nm to 1550 nm. In one preferred embodiment, the first and second coordinates are represented by the Cartesian coordinates x and y, respectively, and each of the non-reentrant surfaces has a two-dimensional sag curve of the form z(x,y)=z(x)+z(y).
In another preferred embodiment of the invention, there is provided an optical system that comprises at least one optical element that includes a first aspherical, non-planar, non-reentrant surface and a second aspherical, non-planar, non-reentrant surface, in which the first and second optical surfaces are aligned along an optical axis and configured to transform substantially all of a substantially non-uniform input beam incident on the first surface to an output beam propagating away from the second surface. The intensity profile of the input beam is expressible as a first function of a first coordinate times a second function of a second coordinate, in which the first and the second coordinates are orthogonal to each other. The intensity profile of the output beam is sigmoidal and has a shape that is substantially different from that of the input beam. The output beam includes a region over which the optical intensity is substantially uniform. This region includes most of the optical power in the output beam, with the intensity of the output beam outside the region varying gradually to substantially reduce diffraction effects. In a preferred embodiment, the optical system comprises first and second optical elements that include the first and the second surfaces, respectively. In preferred embodiments, the surfaces are arranged in either a Keplerian or a Galilean configuration. In one preferred embodiment, the first and the second coordinates are orthogonal spatial coordinates. In a preferred embodiment, the intensity profile of the input beam is symmetric about the optical axis. In one preferred embodiment, substantially all of the input beam is transformed for any wavelength within the wavelength range extending from 257 nm to 1550 nm. In preferred embodiments, the first and second coordinates are represented by the Cartesian coordinates x and y, respectively, and each of the non-reentrant surfaces has a two-dimensional sag curve of the form z(x,y)=z(x)+z(y).
In one aspect of the invention, there is provided a method of transforming a beam of electromagnetic radiation. The method includes providing at least one optical element that includes a first and a second aspherical, non-planar, non-reentrant surface. An input optical beam of substantially non-uniform intensity is directed onto the first surface, in which the intensity profile of the input beam is expressible as a first function of a first coordinate times a second function of a second coordinate, with the first and second coordinates being independent of each other. The method further includes transforming substantially all of the input beam into an output beam that propagates away from the second surface, in which the aspherical surfaces are selected to perform said transforming. The output beam includes a central region over which the optical intensity is substantially uniform, with the central region including most of the optical power in the output beam, and the intensity of the output beam outside the region varies gradually to substantially reduce diffraction effects. In a preferred implementation of the method, said at least one optical element includes a first and a second optical element, with each of the first and the second optical elements having at least one aspherical, non-planar, non-reentrant surface, in which the method further comprises aligning the first and the second optical elements along an optical axis. In one preferred implementation, the first and second coordinates are orthogonal spatial coordinates, and the intensity of the output beam outside the central region varies gradually to substantially reduce diffraction effects.
In another aspect of the invention, there is provided a method of designing an optical system for transforming a first optical beam to a second optical beam, in which the first and second optical beams have respective intensity profiles. The method includes expressing the intensity profile of the first optical beam as a first function times a second function, and expressing the intensity profile of the second optical beam as a third function times a fourth function, in which each of the first and third functions are functions of a first spatial coordinate, and each of the second and fourth functions are functions of a second spatial coordinate, with the first and the second spatial coordinates being orthogonal to each other. The method further includes defining an optical axis along a spatial coordinate orthogonal to the first and second spatial coordinates along which two aspherical, non-planar, non-reentrant surfaces are to be aligned. The method includes constructing a ray-tracing function for the first spatial coordinate using the first and third functions, and constructing a ray-tracing function for the second spatial coordinate using the second and fourth functions. The ray-tracing functions are used to calculate sag values for each of the surfaces, in which each sag value is expressible as a sum of contributions that depend on the first and second spatial coordinates, respectively. In a preferred implementation of the method, the first and the second surfaces form part of first and second optical elements, respectively. The method allows for aspherical, non-planar, non-reentrant, surfaces to be arranged in either a Keplerian or a Galilean configuration. In a preferred implementation, the first optical beam is substantially non-uniform, such as a Gaussian. In one preferred implementation, the intensity profile of the second optical beam is substantially rectangular, and is preferably substantially uniform. The second output beam may advantageously have an intensity distribution selected from the distributions consisting of Fermi-Dirac, super Gaussian, and flattened Gaussian. The integrated intensity of the second output beam may advantageously be at least 90% that of the first optical beam. In a preferred implementation, the method further includes manufacturing the optical system.